Grothendieck's $p$-curvature conjecture for linear differential equations

Let $L\in\mathbb{Q}[x][\partial]$ be a differential operator, and for almost all primes $p$ let $L_p\in\mathbb{F}_p[x][\partial]$ denote its reduction modulo $p$. A basis of solutions of $L_py=0$ is $\mathbb{F}_p[[x^p]]$-linearly independent when its elements are linearly independent over $\mathbb{F}_p[[x^p]]$.

Grothendieck's $p$-curvature conjecture. If $L_py=0$ has a basis of $\mathbb{F}_p[[x^p]]$-linearly independent solutions in $\mathbb{F}_p[[x]]$ for almost all prime numbers $p$, then there exists a basis of $\mathbb{Q}$-linearly independent algebraic solutions of $Ly=0$ in $\mathbb{Q}[[x]]$.

The conjecture characterizes, through reductions modulo almost all primes, differential equations with algebraic power-series solutions. It is known for order-one equations and for Picard–Fuchs equations, with further partial results, but the general case, including order two, remains open.